3.164 \(\int \sqrt{a+b x^2} \sqrt{c+d x^2} \, dx\)

Optimal. Leaf size=249 \[ \frac{2 c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{1}{3} x \sqrt{a+b x^2} \sqrt{c+d x^2}+\frac{x \sqrt{a+b x^2} (a d+b c)}{3 b \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

((b*c + a*d)*x*Sqrt[a + b*x^2])/(3*b*Sqrt[c + d*x^2]) + (x*Sqrt[a + b*x^2]*Sqrt[
c + d*x^2])/3 - (Sqrt[c]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x
)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]
*Sqrt[c + d*x^2]) + (2*c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt
[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c
+ d*x^2])

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Rubi [A]  time = 0.421404, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{2 c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{1}{3} x \sqrt{a+b x^2} \sqrt{c+d x^2}+\frac{x \sqrt{a+b x^2} (a d+b c)}{3 b \sqrt{c+d x^2}}-\frac{\sqrt{c} \sqrt{a+b x^2} (a d+b c) E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{3 b \sqrt{d} \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x^2]*Sqrt[c + d*x^2],x]

[Out]

((b*c + a*d)*x*Sqrt[a + b*x^2])/(3*b*Sqrt[c + d*x^2]) + (x*Sqrt[a + b*x^2]*Sqrt[
c + d*x^2])/3 - (Sqrt[c]*(b*c + a*d)*Sqrt[a + b*x^2]*EllipticE[ArcTan[(Sqrt[d]*x
)/Sqrt[c]], 1 - (b*c)/(a*d)])/(3*b*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]
*Sqrt[c + d*x^2]) + (2*c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt
[c]], 1 - (b*c)/(a*d)])/(3*Sqrt[d]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c
+ d*x^2])

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Rubi in Sympy [A]  time = 57.1388, size = 214, normalized size = 0.86 \[ \frac{2 a^{\frac{3}{2}} \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{3 \sqrt{b} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{x \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{3} - \frac{\sqrt{c} \sqrt{a + b x^{2}} \left (a d + b c\right ) E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{3 b \sqrt{d} \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} + \frac{x \sqrt{a + b x^{2}} \left (a d + b c\right )}{3 b \sqrt{c + d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2),x)

[Out]

2*a**(3/2)*sqrt(c + d*x**2)*elliptic_f(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/
(3*sqrt(b)*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x**2)) + x*sqrt(a +
b*x**2)*sqrt(c + d*x**2)/3 - sqrt(c)*sqrt(a + b*x**2)*(a*d + b*c)*elliptic_e(ata
n(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(3*b*sqrt(d)*sqrt(c*(a + b*x**2)/(a*(c + d*
x**2)))*sqrt(c + d*x**2)) + x*sqrt(a + b*x**2)*(a*d + b*c)/(3*b*sqrt(c + d*x**2)
)

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Mathematica [C]  time = 0.44176, size = 198, normalized size = 0.8 \[ \frac{d x \sqrt{\frac{b}{a}} \left (a+b x^2\right ) \left (c+d x^2\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d-b c) F\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )-i c \sqrt{\frac{b x^2}{a}+1} \sqrt{\frac{d x^2}{c}+1} (a d+b c) E\left (i \sinh ^{-1}\left (\sqrt{\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{3 d \sqrt{\frac{b}{a}} \sqrt{a+b x^2} \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x^2]*Sqrt[c + d*x^2],x]

[Out]

(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2) - I*c*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqr
t[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)] - I*c*(-(b*c) +
a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (
a*d)/(b*c)])/(3*Sqrt[b/a]*d*Sqrt[a + b*x^2]*Sqrt[c + d*x^2])

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Maple [A]  time = 0.016, size = 328, normalized size = 1.3 \[{\frac{1}{ \left ( 3\,bd{x}^{4}+3\,ad{x}^{2}+3\,c{x}^{2}b+3\,ac \right ) d}\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( \sqrt{-{\frac{b}{a}}}{x}^{5}b{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}a{d}^{2}+\sqrt{-{\frac{b}{a}}}{x}^{3}bcd+ac\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) d-\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticF} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) b{c}^{2}+\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) acd+\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{{\frac{d{x}^{2}+c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{{\frac{ad}{bc}}} \right ) b{c}^{2}+\sqrt{-{\frac{b}{a}}}xacd \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2),x)

[Out]

1/3*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*((-b/a)^(1/2)*x^5*b*d^2+(-b/a)^(1/2)*x^3*a*d
^2+(-b/a)^(1/2)*x^3*b*c*d+a*c*((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*EllipticF(
x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*d-((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2)*Ellipt
icF(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c^2+((b*x^2+a)/a)^(1/2)*((d*x^2+c)/c)^(1/2
)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*a*c*d+((b*x^2+a)/a)^(1/2)*((d*x^2+c)
/c)^(1/2)*EllipticE(x*(-b/a)^(1/2),(a*d/b/c)^(1/2))*b*c^2+(-b/a)^(1/2)*x*a*c*d)/
(b*d*x^4+a*d*x^2+b*c*x^2+a*c)/(-b/a)^(1/2)/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c),x, algorithm="fricas")

[Out]

integral(sqrt(b*x^2 + a)*sqrt(d*x^2 + c), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{a + b x^{2}} \sqrt{c + d x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2),x)

[Out]

Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{b x^{2} + a} \sqrt{d x^{2} + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c), x)